How to fix the pulling corners on a smooth surface?
The main problem is that the handle is extruded from the middle of a face instead of from a proper "edge looped" one. Your model ends up with non quads just above the handle and subdiv modifier does not like that.... At this point you might be better off starting again from a cylinder with enough subdivisions for one face to match the size of the handle. Check out the difference in topology between the two models below. The left one started from a 10-face cylinder, the right one from a 20-face cylinder: